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Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.There are two trails near Sandra's house that she runs regularly, a short loop and a long loop. Last week, she ran short loops and long loops, for a total of miles. This week, she ran short loops and long loops, covering a total of miles. What is the length of each loop?The short loop has a length of miles, and the long loop has a length of miles.
Full solution
Q. Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.There are two trails near Sandra's house that she runs regularly, a short loop and a long loop. Last week, she ran short loops and long loops, for a total of miles. This week, she ran short loops and long loops, covering a total of miles. What is the length of each loop?The short loop has a length of miles, and the long loop has a length of miles.
- Define variables: Define the variables for the lengths of the short and long loops.Let be the length of the short loop in miles.Let be the length of the long loop in miles.
- Write equations: Write the system of equations based on the given information.First week: short loops and long loops for a total of miles.Second week: short loops and long loops for a total of miles.
- Eliminate variable: Decide which variable to eliminate.We can eliminate by subtracting the second equation from the first because the coefficients of are the same in both equations.
- Subtract equations: Subtract the second equation from the first to eliminate .
- Solve for y: Solve for y.
- Substitute value: Substitute the value of back into one of the original equations to solve for . Using the second equation:
- Verify solution: Verify the solution by substituting the values of and into the other equation.Using the first equation: The values satisfy the equation, so the solution is correct.
